Searching for Dark Matter at the LHC

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Dark Matter (DM or \(\chi\( ) is a hypothetical form of non-luminous, non-baryonic matter surmised to explain a number of observational inconsistencies stemming from various large astrophysical systems. The search for Dark Matter is an ongoing area of research in physics and astronomy. The nature and properties of DM remain largely enigmatic, with proposed DM candidates in literature spanning tens of orders of magnitude in mass, from particles infinitesimally lighter than electrons to massive primordial black holes [REF]

Opening remarks: the WIMP Dark Matter landscape

In the twentieth century, advancements in high-energy physics significantly enhanced our understanding of the fundamental nature of the universe, its matter composition, and the interactions therein. This success culminated in the early twenty-first century when the Large Hadron Collider experiments discovered the Higgs boson, effectively completing the particle puzzle within the framework of the Standard Model (SM) of particle physics. Despite this success, astrophysical observations [hep-ph/0404175] have presented compelling evidence, assuming the validity of known gravitational laws and general relativity theory, for the existence of an invisible, yet unexplained, component of matter. These observations, spanning from the rotational velocities of galaxies to precise measurements of the cosmic microwave background and gravitational lensing, consistently support the presence of an invisible matter component. This matter, not accounted for by the Standard Model, is five time more abundant than regular matter.

The interactions and properties of DM remain largely unknown, as the only recognized interaction attributed to dark matter is gravitational, underscoring the subtlety and elusive nature of this mysterious cosmic constituent. Proposed candidates in literature span vast orders of magnitude in mass, ranging from elementary particles infinitesimally lighter than electrons to sizable primordial black holes. Among these candidates, a particularly compelling and extensively studied contender is a novel class of subatomic particles known as weakly interactive massive particles (WIMPs). WIMPs currently stand as the prevailing paradigm in the pursuit of particle dark matter and is the focus of this article. For further reading beyond the WIMP paradigm, we refer the reader to [REF].

WIMPs’ masses range between 1-100 times that of the proton. WIMPs exhibit only weak-scale interactions with ordinary Standard Model particles. In the standard WIMP scenario dark matter is made of one single type of particles, and the new physics particles are protected by a discrete symmetry so that new particles are always produced in pair. These elusive particles originated in the early stages of the universe and persist today, filling it with a precisely measured density commonly referred to as relic density, relic abundance, or Ω. This relic density is derived from the fundamental principles of the Standard Model of cosmology, also known as the Big Bang Theory, and an effect known as freeze-out. In the early universe, highly energetic and massive particles were created and existed in thermal equilibrium, facilitated by mechanisms like pair production or collisions/interactions with other particles. Essentially, processes converting heavy particles into lighter ones and vice versa occurred at the same rate. However, as the universe expanded and cooled, two significant changes took place: 1) lighter particles lost the necessary kinetic energy (thermal energy) to generate heavier particles through interactions, and 2) the expansion of the universe reduced the particle density, leading to less frequent or entirely absent interactions.

At a certain point in the universe's expansion, the density of heavier particles or a specific particle species became too low to sustain regular interactions, breaking thermal equilibrium. This phenomenon is described as "freeze-out" or "decoupling" and the number of particles, no longer influenced by interactions, remains constant thereafter. The precise moment or temperature of freeze-out can be calculated by equating the particle interaction rate with the Hubble (expansion) rate of the Universe. The density of a particular particle at the time of freeze-out is referred to as the relic density for that particle, as its abundance remains constant.

The significance of WIMPs in the context of dark matter is further strengthened by what is often termed the "WIMP miracle." This concept highlights a fortunate coincidence which aligns the predicted abundance of WIMPs, based on their weak-scale interactions and mass range, to the observed cosmic dark matter density. Remarkably, the weak force's interaction strength and the mass scale of WIMPs naturally lead to a relic density that closely matches the universe's dark matter density. This coincidence suggests that WIMPs, through their freeze-out process, provide just the right amount of dark matter to account for the gravitational effects observed in galaxies and galaxy clusters, without necessitating fine-tuning of their properties. The WIMP miracle has been often considered a strong supporting factor to the search for DM WIMPs as this elegant convergence of theoretical prediction and observational evidence highlights the potential of WIMPs to bridge our understanding of particle physics with the cosmological phenomena that shape the universe.

Figure 1: Schematic representation of the complementary experimental approaches to the detection of dark matter.

The scientific quest to understand WIMP DM is a multi-disciplinary effort involving different and complementary experimental techniques, briefly outlined in the following and depicted in Figure 1.

  • Direct detection searches - This methodology attempts to detect the interaction (elastic scattering) of DM off nuclei using specialised low-background detectors. This search method relies on the assumption that DM is distributed as a halo in the Milky Way and the Earth is consequently exposed to a constant high flux of DM particles.
  • Indirect detection searches - This methodology attempts to detect the annihilation or DM annihilations in astrophysical systems, where the DM density is at its highest. Various ground and space-based specialised instruments aim to measure the flux of such DM annihilation or decays products.
  • Particle collider searches - In high-energy accelerators, particles that were present in the early universe can be recreated and studied in detail. This methodology aims to discover and analyse DM particles produced in a controlled laboratory environment.

The Grimoire of the Collider Experimentalists

How to detect the invisible

If dark matter particles exhibit non-gravitational interactions with Standard Model matter, their production could occur in collisions generated by particle accelerators. Currently, the Large Hadron Collider (LHC) stands as the highest-energy particle collider, operational at CERN in Geneva, Switzerland, and has been collecting data since 2009. Operational at four interaction points along the LHC circumference, the two beams of protons are brought into collision. Two general-purpose experiments, namely ATLAS [add link to scholarpedia article] and CMS, along with two specialized experiments (LHCb and ALICE), collect the outcomes of these collisions. In the following discussion, our focus will be on the two general-purpose detectors, ATLAS and CMS, whose experimental agenda encompasses a broad spectrum of searches for potential DM particle candidates and their associated interactions. For further in-depth study of the DM search program of LHCb (and Belle-II) expertiments, we refer the interested reader to [REF].

It is important to highlight that if DM particles were to be produced in a LHC collision, these particles would be produced in pairs and pass through the detector without leaving any signal, due to their extremely weak interaction cross section with the ordinary SM particles of the detector. These "invisible" signatures, depicted in Figure 2., cannot be triggered or measured in the typical environment of a hadronic collision, characterised by extreme pile-up and underlying event conditions Figure 3.. Dark Matter particles are measured instead using momentum conservation. Collider experiments search for events where DM is produced in conjunction (association) with ordinary SM particles. These SM particles are generated either through a series of decays that culminate in the production of the DM particle or as part of the initial state radiation. Since the proton momenta in the plane perpendicular to the beam axis is zero before the proton-proton collisions, due to momentum conservation the total transverse momentum of the final state has to be zero after the collision as well. Therefore, the DM and SM particles systems will be produced back-to-back in this plane. Since the DM particle is “invisible” and the SM particle is not, this characteristic momentum-imbalanced or missing energy signature (ETmiss) is a clean fingerprint for DM Figure 4..

Figure 2: A representation of a DM event in the plane transverse to the proton-proton collision
Figure 3: An example of the typical Pile-Up conditions of a proton-proton collision event
Figure 4: Event display of a typical "Monojet" event, where invisibles are produced in association with an energetic jet.

The theoretical codex

Searches for DM at colliders need a theoretical framework to predict the DM production mechanism. The design and interpretation of every search depends on these theoretical assumptions. The vast majority of collider DM searches assume that the DM candidate is a Dirac (or sometimes Majorana) fermion. The DM production mechanism is determined by the theoretical assumptions determining the interaction of DM with the SM. This means that strictly speaking we do not only need to postulate a dark matter particle, but also a way for this state to communicate to the SM. Therefore, frameworks which provide a theoretical description for DM and its interaction will postulate (with minor exceptions) the existence of at least two new particles.

Theoretical frameworks used for DM searches can have different degrees of complexity and, in turn different levels of theoretical completeness. In general, the more theoretically complete is a theory, the more complex, in terms of parameter dependence is its phenomenology and particle spectrum (Fig 4). One of the most well-known examples of a complete Beyond the Standard Model theory is Supersymmetry [REF]. Such model cannot be used to guide experimental searches out-of-the box, as the underlying phenomenology depends on a parameter space with too many dimensions. On the other side of the completeness-complexity spectrum is the Effective Field Theory approach, which aims to describe new physics phenomena in terms of few interactions’ operator and one interaction scale. This theoretical framework has been widely used to search for DM during Run 1 of the LHC operations, using collisions recorded at 8 TeV centre-of-mass energy. This approach suffered heavy limitations already with Run-1 data and could not be used to analyse and interpret collision data recorded during Run 2 of the LHC data taking. In fact, the high centre-of-mass energy of the LHC collisions breaks the validity of the effective field theory regime, as the momentum transverse of the collision Q^2 exceeded in a large fraction of the events the mass of the new mediator [REF].

The theoretical framework employed for all DM collider searches during Run 2 represents a pragmatic balance between the theoretical comprehensiveness inherent in complete Beyond the Standard Model theories and the simplicity afforded by single-parameter Effective Field Theories. This category of models is conventionally denoted in the literature as Simplified models. In this discussion, we will also introduce UV-complete and renormalizable extension of these Simplified models, termed Minimal models (Figure 5 ), which predict more than one new particle. It is worth noting that, although the distinction between the two classes of models is made in this article, it is common sometimes in the literature to treat them without differentiation.

Figure 5: A representation of the complexity of DM models at collider, the natural compromise between theoretical completeness (e.g. renormalizability), complexity (number of new particles) and generality of the categories of models discussed in this review.

To date, the largest fraction of collider DM searches can be grouped into two broad categories of models, which can be distinguished by the fundamental properties of the new interaction.

In the first category of models, which are termed Supersymmetry-inspired models (SUSY) in this review, the new particles carry some of the SM charges or quantum numbers and the new interaction act as intermediaries between the SM particles and the new physics sector (Figure 5 left). SUSY is one of the main foci of the experimental search program of the LHC since the beginning. It is a theory grounded on a generalisation of space-time transformation linking fermions and bosons and predicting a zoo of new particles, each the supersymmetric partner of one of the known SM particles. The lightest supersymmetric particle, the neutralino in most models, is stable and the DM candidate of these models. As the search for neutralinos is closely entangled to the search for SUSY particles in general, it is not discussed further in this article and we refer to [add reference scholarpedia article] for further discussion.

In the second category of models, termed Mediator-DM models in the following but also referred to as hidden sector, dark sector or portal models in the literature, may not carry Standard Model charges or quantum numbers, or if they do, they interact in a way that typically involves only new particles or indirect interactions with the Standard Model (Figure 5 center). These models often introduce hidden sectors or dark sectors that are only weakly coupled to the SM, usually through higher-order processes or via specific mediator particles. In the latter case, DM-SM interaction is mediated by a new particle, which mediates the interaction (force) between the dark sector to which the DM belongs and the ordinary SM sector. The mediator is a charge neutral state of unknown mass and width that can fulfil various spin hypothesis. The model is characterised by a minimal number of free parameters: The mediator mass and decay width, the mediator coupling to the SM and the DM particles (gSM and gDM in Fig. Figure 5) and the mass of the DM particle. Different parameter choices, in particular different spin assumptions for the mediator, give rise to different collider phenomenology, or in other terms different possible signatures in the detector. Under all parameter assumptions, the mediator can decay either in a pair of DM particles or in a pair of SM particles (Figure 5 right). For this reason, also searches for new resonances play an important role in the investigation and constraint of mediator-DM models.

Figure 6: Representative Feynmann-like dyagrams for collider production of (left) a typical SUSY interaction, (center and right) typical DM-mediator interactions with or without DM particles in the final state

Minimal models belonging to this class feature, in addition to the DM-mediator pair, extended Higgs or gauge sectors, which allow the introduction of model characteristics that make them theoretically sounds in terms of renormalisation and UV-completion. The most studies cases are:

- two-Higgs-doublet models (2HDM) [1] - two-mediator-dark-matter models (2MDM) [2]

Also in the case of Minimal DM models, the choice of parameters and properties of the DM-mediator pair as well as the particles that compose the extended SM sector define the collider phenomenology and the specific detector signature to be investigated to probe these models. Also in this case, resonant searches for the new particles of the extended SM sector play an important role in probing these models.

The dependence on the theoretical model is a peculiar trait of the collider searches for DM. However, despite being theoretically inspired, these searches are experimentally designed to be as model-independent as possible. Even more importantly these searches are sensitive to any DM signal which may give rise to an excess of data with respect to the SM expectation in the target phase space of each DM search. While the exact magnitude of the excess is model dependent, the searches are conducted such that they remain sensitive to any kind of new that enters the search phase space. The theoretical framework is also used for the interpretation of the results, in order to extract information about the nature of DM particles from a non-discovery. This information is conventionally expressed in terms of exclusion limits on selected DM benchmark models, which may also be used to compare with direct and indirect DM searches.

Results: status quo on collider Dark Matter

To date, the ATLAS and CMS experiments have published over 50 analyses sensitive to DM production at colliders. None of these searches has found a significant excess in the data over the expected SM background and therefore each has been interpreted in exclusion limits for DM models under selected parameter assumptions. In the subsequent sections, we offer a comprehensive review of the parameter space eliminated by experimental searches conducted at colliders by the conclusion of the LHC Run-2. The selection of models for presenting these findings is guided by two main objectives: firstly, to furnish a broad overview of signature coverage, and secondly, to prioritize models that have received considerable attention. For each model considered, we delineate the primary final states of interest and highlight results within specific parameter space regions. We focus on two simplified and two minimal mediator-dark matter (DM) models to encapsulate a diverse spectrum of searches and final states.

Spin-1 Mediator (simplified model)

Model in a nutshell: a simple extension of the SM with an additional U(1) gauge symmetry under which the DM particles are charged. The new mediator (Z') is either a vector or an axial-vector boson. The mediator interaction is parametrized by one coupling to DM particles, one flavor-universal couplings to quark and, in some cases, also a non-vanishing flavor-universal coupling to leptons.

Final states:

  * ETmiss plus X, X = (ISR) jets / Vector Bosons (W/Z) / photons (g) 
  * Resonant pairs of leptons / jets / b-quarks / top quarks. 

Results are conventionally presented on a parameter plane which scans both the mediator and the DM mass. The coupling assumptions are fixed to constant values as highlighted on the plots. Figure 6 gives an example of exclusion reach from the CMS and ATLAS Experiments for a (left) vector and (right) axial vector mediator and different coupling assumptions. It is noteworthy that in all cases, the reach of the ETmiss plus X signatures is capped from above by the kinematic decay threshold of the mediator into a pair of dark matter particles (m(Z') = 2 m(DM)), and it is restricted by the model production cross-section for high masses. Conversely, the reach of searches for resonant pairs of fermions is consistently independent of the dark matter mass, thus uniquely probing scenarios where the direct decay of the mediator into DM particles is kinematically prohibited. The efficacy of these searches is constrained at low masses by the momentum thresholds of the ATLAS and CMS triggering systems, and at high masses, by the production cross-sections of these models, hence the dataset luminosity.

Figure 7: example of exclusion reach from the CMS and ATLAS Experiments for a (left) vector and (right) axial vector mediator and different coupling assumptions.

In depth: Comparison with Direct Detection searches

Figure 8: Comparison of exclusion reach in terms of nucleon scattering cross section between collider DM searches and direct detection DM searches.

As non-collider and collider DM searches probe unique aspects of DM existence and properties, it is interesting and informative to compare the sensitivity and current reach of such complementary methods in a somewhat coherent way. This is conventionally achieved (see as an example Fig 6 for spin-1 simplified models) by translating the LHC results onto the non-collider interpretation planes (nucleon scattering cross-section versus DM mass). This procedure avoids difficulties associated with the fact that while direct and indirect detection bounds are valid for multiple DM models, the LHC limits hold true only for the specific mediator model and parameters used in the interpretation. The exclusion contours can be viewed as a transposition of the results presented in Fig. 7, where the resonant searches exhibit again their characteristic independence on the DM mass, now shown in the x axis of the plot and the ETmiss plus X searches exhibit again a triangular end-point which is a complex function of the mediator mass, SM-mediator coupling (which affects the scattering cross section) and the kinematic threshold m(Z') = 2 m(DM).

Spin-0 Mediator (simplified model)

Model in a nutshell: a new spin-0 gauge particle (scalar or pseudoscalar) mediates interactions with DM particles. There are four parameters: mediator mass, dark matter mass, DM-mediator coupling (g), and mediator coupling with standard model fermions. The latter is the product of a flavor-universal term (gq) with the SM-Yukawa coupling for each fermion. This interaction pattern, known as the minimal flavor violation ansatz, relaxes constraints on the coupling of new spin-0, color-neutral particles to SM fermions imposed by flavor measurements. It notably also implies that these mediators are produced through loop-induced gluon fusion or with heavy-flavor quarks.

Final states: ETmiss plus X, X = (ISR) jets / top quark pairs / single top quarks / tW / b-quark pairs Resonant pairs of b-quarks / top quarks; Resonant pairs of top quarks in association with top quark pairs (4 tops).

Results (shown for the scalar mediator model in Fig 8 for both ATLAS and CMS experiments) are presented, for this model, as a function of mediator mass and excluded signal strength. The latter is expressed as the ratio of the excluded cross section to the nominal model cross section for the specific choice of unitary couplings. The mass of the DM is set to 1 GeV but the results are valid for all DM masses for which the mediator decay to \chi\chi kinematic threshold is open. The presentation of the results in such fashion is a relic of the fact that only ETmiss plus X searches have been interpreted for this model. Additional complications, related to the scalar nature of the mediator, present a non-negligible interference between the mediator and SM processes when the former decays into SM final states and dedicated searches that account for these affects have been published to date only for the 8 TeV dataset [REF]. The most sensitive final state by far is the ETmiss + tt final state, which is a statistical combination of specific analyses selecting each one of the three most sensitive decays of the top quark pair: all-hadronic, single lepton and di-leptonic final states. The ETmiss + bb channel is the least sensitive in the parameter plan where results are presented only because this simplified model lacks additional regulatory parameters that can naturally enhance the mediator to bottom quark couplings over the top quark ones. These parameters are introduced in minimal model such as the 2HDM+a (see next section).

The hypothesis that the SM Higgs boson can mediate the interaction between the SM and DM particles is a specific realization of spin-0 mediator models. In the SM, the invisible Higgs boson branching ratio, B𝐻→inv, is 0.12% from 𝐻 → 𝑍𝑍 → 4𝜈 decays, and higher branching ratios to invisible particles are predicted by Higgs–dark-matter mediator models. Experimental constrains on the Higgs invisible branching ratio have approached the 10% mark at the end of the LHC Run2 [REF].

Figure 9: <add caption>

Spin-0 mediator (minimal model)

Model in a nutshell: a pseudoscalar mediator simplified model is embedded into a UV-complete and renormalizable framework by involving a type-II two-higgs-doublet extension of the Higgs sector [REF]. The alignment limit is assumed, so that the lightest CP-even state, h, of the Higgs sector can be identified with the SM Higgs boson. The model is characterised by one DM mediator, one DM particle, two heavy scalar Higgs partners (A, H) and one charged Higgs pair (H^\pm). The most notable free parameter of the model, out of a total of fourteen including the masses of all new particles, are the mixing angle between the CP-odd Higgs partner and the pseudoscalar DM mediator, sinq, and the ratio of the vacuum expectation value of the two Higgs doubles, tanb. These two parameters strongly affect the phenomenology and the relative production cross section of different final states.

Final states: ETmiss plus X, X = SM Higgs, Z boson, tW (dominant production modes) ETmiss plus X, X = (ISR) jets / top quark pairs / b-quark pairs (sub-dominant processes) Resonant pairs of leptons / b-quarks / topquark / top-b quark pair (decay modes of Higgs partners) Resonant pairs of top/b quarks in association with top/b quark pairs (4 top quarks/4 b-quarks/2 b-quarks+2 top quarks).

Results are presented in various selections of parameter planes, in an attempt to characterise the complementarity of phenomenology as a function of the most relevant parameters of the model. We show, as an example, in Fig 9 the parameter plane scanning the mass of the CP-odd Higgs partner, A, and tanb. The dominant sensitivity to this model is provided by searches targeting the resonant production of the DM pseudoscalar mediator via one of the charged or neutral Higgs partners, which gives rise to the following signatures: ETmiss plus SM Higgs (H⟶ah), ETmiss plus Z (A⟶aZ) and ETmiss plus tW (t H^\pm⟶t aW). In the benchmark, the masses of the Higgs partners are always set to equal values. While this assumption is an easy solution to ensure that the model fulfills electroweak and flavour constraints [REF], it doesn't have to be realised so strictly. Consequently, while the sensitivity coverage of the three dominant searches is highly overlapping in Fig. 9 due to this assumption, the three searches are highly complementary in nature. Resonant searches for tbH^\pm and ttA/H also provide complementary sensitivity to this model. sub-dominant signatures and additional resonant final states have yet to be interpreted for this model by the experimental collaborations.

As final word, we bring attention to the reader to the fact that DM models, especially ones with a broad and parameter dependent phenomenology such as the 2HDM+a, are indirectly constrained also by the extensive program of precision SM measurements performed at collider experiments. An example of such usage is presented In Fig. 10 [REF], where a considerable reach is obtained using SM unfolded measurements. The dominant sensitivity arises from measurements targeting processes with W-bosons in the final state (di-bosons or top quark pair production) in final states with all jets or exactly one lepton.

Figure 10: <add caption>

In depth: Relic density considerations

Figure 11: <add caption>

The capacity of DM collider models to accurately forecast the observed relic density holds significant implications within the realm of cosmology. The relic density is sensitive to the properties of of the dark matter particle, including its interactions with SM particles and itself — fundamental properties upon which collider searches rely. Additional theoretical mechanisms almost always permit adjustments to the relic density prediction within a model. Consequently, while the relic density serves as a guiding principle, it is not inherently a constraint on the parameter space in DM collider searches. As an instructive example, Figure 7 compares the predicted relic density for a 2HDM+a model against the present exclusion limits established by various experimental searches. It generally holds true across broad model categories that very light DM candidates often yield relic density predictions that exceed the observed value (overabundant prediction). These predictions also frequently intersect with measured values and exhibit significant underproduction in close proximity to kinematic thresholds, such as m(a) = 2 m(DM) in the Figure. ETmiss plus X searches sensitivity is largely independent of the specific DM mass, once below such kinematic thresholds. This is realized thanks to inherent sensitivity os such searches to the mediator transverse momentum rather than the DM kinematics. From these considerations, one can reliably argue that such searches probe a parameter space with noteworthy cosmological implications. Once the mediator-DM decay mode is kinematically forbidden, the model frequently results in an underproduction of relic density, as observed in the aforementioned example. This portion of parameter space is complementarily explored by mediator resonant searches. The significance of this exploration from the cosmological standpoint, in terms of relic density prediction, is contingent upon the specific model and necessitates assessment on a case-by-case basis.

Dark Higgs mediator (minimal model)

Model in a nutshell: the spin-1 vector simplified model (Z') is extended by an additional complex dark Higgs field which is responsible for generating the DM and the Z' masses through Yukawa interactions. All three new particles the Z', the DM and the dark Higgs s couple to each other’s. Non-zero mixing between the dark Higgs boson and the SM Higgs boson ensures that the dark Higgs boson is unstable even if it is the lightest state in the dark sector and decays into SM final states (similar to the SM Higgs) with a negligible lifetime.

Final states: Searches for Z': ETmiss plus (ISR) jets Searches for dark-Higgs-strahlung (from the Z'): ETmiss plus s(bb) / s(WW) / s(ZZ)

Results are usually presented in the parameter plan which scans the mass of the Z' and the mass of the dark Higgs s. As an example, searches targeting the WW decays of the dark Higgs are presented in Fig 12. The sensitivity of these searches is limited in s mass range by the s ➞WW kinematic decay threshold. The sensitivity has a lower limit in Z' mass due to the specific choice of DM mass for the presented result and it is limited by the production cross section of the Z' boson for high Z' masses.

Figure 12: <add caption>

Final words

The significance of delving into the intricacies of specialized terminology, tracing the historical evolution, and understanding the diverse approaches within the field of collider dark matter searches cannot be overstated. The comprehensive definition of this topic has been a product of robust collaboration across multinational experimental collaborations and diverse scientific communities. For further insights into the literature, in form of whitepapers and recommendations, of this collaborative endeavor and its historical context, interested readers are encouraged to explore the resources available within the LPCC DM Working group: - <link>


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